2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74195We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-handlebody is homotopic to a complex one.24 pages, 9 figuresGeometric TopologyDifferential Geometry57M20; 32C60text